Mathematical Sciences: RUI: Spectral Problems for Toeplitz and Other Structured Matrices

一个方阵族H是有效结构的，如果 指定所需的独立参数的数量 H的任意n乘n个成员近似正比于n 当n很大时。 Toeplitz、Hankel和Toeplitz + Hankel 矩阵是高效结构化矩阵的示例， 给定位移秩的矩阵，如T所定义。凯拉特 和他的合著者拟议研究的目的是（a） 开发快速算法， 一般有效类的特征值-特征向量对 结构厄米特矩阵，类似于主要的 Toeplitz和Toeplitz + Hankel研究者算法 矩阵;（B）研究Toeplitz矩阵的谱性质 （特别是，偶谱和奇谱之间的关系， 真实的对称Toeplitz矩阵）;（c）研究 设计快速算法来寻找真实的 非对称Toeplitz矩阵的特征值;以及（d）至 考虑Toeplitz-Sturm矩阵的谱性质。 有效结构的埃尔米特矩阵出现在许多 重要的物理和科学应用，包括 涉及预测a未来值的统计问题 随机变量基于其过去值的观察，信号 处理（从数据中提取信息的科学） 发射信号），以及地球物理应用，包括 研究地球的自由振荡和地震现象 比如地震。 在研究这些问题时， 需要确定的个别特征值 非常高阶的有效结构矩阵（在 千）。 拟议研究的主要目标之一是 是发展数值方法来找到这些特征值， 方法的复杂性（计算要求）是 与矩阵阶数的二次幂成比例， 而不是秩序的三次方，因为它是为了 标准的计算方法，不利用 矩阵的特殊结构。

A family H of square matrices is efficiently structured if the number of independent parameters required to specify an arbitrary n by n member of H is approximately proportional to n when n is large. Toeplitz, Hankel, and Toeplitz-plus-Hankel matrices are examples of efficiently structured matrices, as are matrices of a given displacement rank, as defined by T. Kailath and his coauthors. The purposes of the proposed research are (a) to develop fast algorithms for finding individual eigenvalue-eigenvector pairs for general classes of efficiently structured Hermitian matrices, analogous to the principal investigator's algorithms for Toeplitz and Toeplitz-plus-Hankel matrices; (b) to study spectral properties of Toeplitz matrices (especially, the relationship between the even and odd spectra of real symmetric Toeplitz matrices); (c) to investigate the possibility of devising fast algorithms for finding the real eigenvalues of nonsymmetric Toeplitz matrices; and (d) to consider the spectral properties of Toeplitz-Sturm matrices. Efficiently structured Hermitian matrices arise in many important physical and scientific applications, including statistical problems involving prediction of future values of a random variable based on observations of its past values, signal processing (the science of extracting information from transmitted signals), and geophysical applications including the study of free oscillations of the Earth and seismic phenomena such as earthquakes. In studying these problems it is often necessary to determine individual eigenvalues of efficiently structured matrices of very high order (in the thousands). One of the main objectives of the proposed research is to develop numerical methods for finding these eigenvalues by methods whose complexity (computational requirements) is proportional to the second power of the order of the matrix, rather than to the third power of the order, as it is for standard computational methods that do not take advantage of the special structure of the matrix.